I don't really know how to do this and I have to explain to my class how I got my answer. Help me!!!

Using Snell’s law ( n1 sin θi = n2 sin θr ) at point A gives

1)        sin i1 = n sin r1             

2)        i1 = n r1

3)        i1 = α1 + β1

4)        r1 = β1 - γ

5)        α1 + β1 = n(β1 - γ)

6)        sin i2 = n sin r2

7)        i2 = n r2

8)        i2 = α2 + β2

9)        r2 = β2 + γ

10)      α2 + β2 = n(β2 + γ)

11)      α1 + α2 = (n - 1)(β1 + β2)

The lens maker's equation is derived by considering the image formed by the first refracting surface (left surface) and then using this image as the object for the second refracting surface. The equation takes into account the radii of curvature and the refractive indices of the lens and the surrounding medium.

The lens maker's equation is derived by considering the image formed by the first refracting surface (left surface) and then using this image as the object for the second refracting surface.

The equation takes into account the radii of curvature and the refractive indices of the lens and the surrounding medium.

The lens maker's equation can be stated as: 1/f = (n₂ - n₁)((1/R₁) - (1/R₂)), where f is the focal length, n₂ is the refractive index of the lens, n₁ is the refractive index of the surrounding medium (usually air), R₁ is the radius of curvature of the first surface of the lens, and R₂ is the radius of curvature of the second surface of the lens.

The standard deviation for the sample is approximately 0.348 hours.

The critical value for a 90% confidence interval is approximately 1.645, and for a 95% confidence interval, it is approximately 1.96.

To calculate the standard deviation for the sample, we use the formula:

[tex]\[ \text{Standard deviation for sample} = \frac{\text{Population standard deviation}}{\sqrt{\text{Sample size}}} \][/tex]

Given the population standard deviation of 2.2 hours and a sample size of 40, we plug these values into the formula:

[tex]\[ \text{Standard deviation for sample} = \frac{2.2}{\sqrt{40}} \][/tex]

[tex]\[ \text{Standard deviation for sample} \approx \frac{2.2}{6.324} \approx 0.348 \text{ hours} \][/tex]

So, the standard deviation for this sample is approximately 0.348 hours.

For the critical values of confidence intervals, we refer to the z-table or use a calculator.

For a 90% confidence interval, the critical value is approximately 1.645, and for a 95% confidence interval, it is approximately 1.96.

These values represent the number of standard deviations from the mean that encompass the specified confidence level.

An X-intercept in a rational function is the point where the graph intersects the x-axis. It is derived by setting the function equal to zero and then solving for the variable x. This solution, x, represents the x-intercept of the rational function.

In Mathematics, the x-intercept of a rational function is the point where the function intersects the x-axis. The x-intercept(s) can be found by setting the function equal to zero and solving for x. When graphing, this point determines where the line crosses the x-axis.

The same principle is applied to the calculation of the x-intercept as other functions. The given rational function is set equal to zero, and then algebraic methods are used to solve for the value of x. For example, if given the rational function y = 2x/3x + 1, setting this function to zero would read as 0 = 2x/(3x + 1). Cross-multiplication can then be applied to find the solution for x, which represents the x-intercept.

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Answer: Which fraction below represents the following statement?“In the third game of the season, Martha made one out of five free throws.” The answer for this questions would be 1/5

Step-by-step explanation: I got it correct on Edge.

The fraction that represents the statement "in the third game of the season Martha made one of 5 free throws is 1/5.

A fraction is written with a numerator and a denominator where the numerator is less than the denominator.

Example: 1/3 is a fraction.

We have,

Total number of free throws = 5

Number of free throws Martha made = 1

The number of free throws Martha made can be written in fraction form as:

= Number of free throws Martha made / Total number of free throws

= 1 / 5

Thus,

The fraction that represents the statement "in the third game of the season Martha made one of 5 free throws is 1/5.

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Final answer:

The rate of change for the chef's cooking scenario is calculated by dividing the lbs of chicken by the number of people for each case. In both situations, the rate of change is 1/4 lb per person.

Explanation:

The question asks to find the rate of change for a situation where a chef cooks a certain amount of chicken for a specific number of people. To calculate the rate of change, we need to divide the amount of chicken by the number of people for each situation, and then compare.

For 9 lbs of chicken for 36 people, the rate is 9 lbs ÷ 36 people = 0.25 lbs/person, which is equal to 1/4 lb per person. For 17 lbs of chicken for 68 people, the rate is 17 lbs ÷ 68 people = 0.25 lbs/person, which is also 1/4 lb per person.

Since the rate of change is the same in both cases, 1/4 lb per person, the correct answer is:

C. 1/4 lb per person

the answer is C (8,20)

Answer:

c) (8, 20)

Step-by-step explanation:

We must plug in the given value of x in the function and confirm that the value of y that is returned to us is the correct one.

a) (0, 18)

[tex]x=0,y=18[/tex]

[tex]y = 16 + 0.5(0)=16+0=16[/tex]

the values for y do not match, so it is not a solution.

b) (5, 19.5)

[tex]x=5,y=19.5[/tex]

[tex]y = 16 + 0.5(5)=16+2.5=18.5[/tex]

again, the values for y do not match so it is not a solution.

c) (8, 20)

[tex]x=8,y=20[/tex]

[tex]y = 16 + 0.5(8)=16+4=20[/tex]

For this option, the values of y do match, so (8, 20) are the missing values.

The value of x in the equation 6 - x = 2(x - 6) is 6. To convert this to feet, the calculation shows that 6 miles is equivalent to 31,680 feet.

To solve the equation 6 - x = 2(x - 6), follow these steps:

So, the value of x is 6.

To convert 6 miles to feet, using the equation 1 mile/5280 feet = 6 miles/x feet, we cross multiply to get:

Therefore, 6 miles is equivalent to 31,680 feet.

True, the period of y = cos x occurs every 2pi units.

Equations using trigonometric functions that hold true for all possible values of the variables are known as trigonometric identities.

Given:

Trigonometric function,

y = cosx.

To find the period:

Use formula,

period = 2π/the coefficient of cosx.

Period = 2π

Therefore, the period of y = cosx is 2π.

To learn more about the trigonometric identities;

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Answer:

a) [-1, ∞-) and [1, ∞+)

b)  (√2/2,∞)  and (0,-√2/2)

c) y=0, y=2 and y=4

Step-by-step explanation:

a) On what intervals is f increasing?  

We need to proceed some steps, to answer it properly. What are the critical points? So  

1) Differentiate the original function

[tex]f(x)=3x^{5}-5x^3+2\\ f'(x)=15x^4-15x^{2}[/tex]

2) Turn this into an equation  

[tex]15x^4-15x^2=0\\ u=x^{2} and \\u^{2}=x^{4} \\ Then\\ 15u^{2} -15u=0\\ \\15u(u^2 -1)=0\\Therefore\\u=0,u=1[/tex]

Substitute back to x

[tex]x^{2} =u\\ x^{2} =u^4\\ x^{2}=1\\ x= 1,x=-1\\ x^{2} =0\\ x=0[/tex]

Those are x-coordinates of the critical points of [tex]f(x)=3x^{5}-5x^3+2[/tex]

Using the critical points x, to find y-coordinates of those critical points: namely, maximum point, saddle point and minimum point

x=-1 [tex]y=3x^{5}-5x^3+2[/tex]

[tex]y=3(-1)^{5}-5(-1)^3+2[/tex]

y=4

(-1,4) Maximum point

--

x=0 [tex]y=3x^{5}-5x^3+2[/tex]

[tex]y=3(0)^{5}-5(0)^3+2[/tex] y=2

(0,2) Saddle Point

--

x=1 [tex]y=3x^{5}-5x^3+2[/tex]

[tex]y=3(1)^{5}-5(1)^3+2[/tex]  

(1,0) Minimum point

f(x) increases when any value of  x ∈ (-∞, -1] and x ∈ [1, ∞+)  is plugged in the function.  Or we can rewrite as  x[tex]x\leq -1\\ \\x\geq 1[/tex]

-------------------------------------

b)On what intervals is the graph of f concave upward?

To find out which intervals are these, we need to calculate the 2nd derivative.

[tex]f(x)=3x^{5}-5x^3+2\\ f(x)'=15x^4-15x^{2} \\f(x)''=60x^3-30x\\[/tex]

Since we want the concave upward

f(x)''>0

Then:

[tex]60x^3-30x>0\\ 30x(2x^2-1)>0\\[/tex]

Rewriting

[tex]30x(\sqrt{2x}-1)[/tex]

Then

x=0, x=[tex]\frac{\sqrt{2}}{2} \\-\frac{\sqrt{2}}{2}[/tex]

Studying the sign

The intervals >0, when the graph is concave upwards.

are (√2/2,∞)  and (0,-√2/2)

c) Write the equation of each horizontal tangent line to the graph of f.

Generally, every time somebody ask for a tangent line of any given function, they provide an information. A specific point.

But when they do not? Like in this case?

We have to look for where the 1st function's derivative is zero.

As we 've done previously in letter a. x=-1, x=0, x=1

Let's plug them back to the original function

[tex]y=3x^{5}-5x^{3}+2\\   y=3(-1)^{5}-5(-1)^{3}+2\\ y=-3-5+2\\ y=-8+2\\ y=4\\ \\y=3x^{5}-5x^{3}+2\\ y=3(0)^{5}-5(0)^{3}+2\\ y=2\\\\ y=3x^{5}-5x^{3}+2\\ y=3(1)^{5}-5(1)^{3}+2\\ y=0[/tex]

So each horizontal tangent line, tangents the maximum, saddle and the minimum point.

Yolanda paid for her movie ticket using 28 coins, all nickels and quarters. The ticket cost $4.

Let n be the number of nickels

and q be the number of quarters

1 nickels = $0.05 , so n nickels cost = 0.05n

1 quarter = $0.25, so q quarters cost = 0.25q

Total number of coins = 28

number of nickels + number of quarters = 28

n + q = 28 ------- equation 1

Total ticket cost =$4

cost of n nickels + cost of q quarters = $4

0.05n + 0.25q = 4 -------> equation 2

system of linear equations can be used to find the number of nickels, n, and the number of quarters, q, Yolanda used are

n + q = 28

0.05n + 0.25q = 4

Final answer:

To solve for final velocity (Vf) in acceleration calculations, identify the initial velocity (Vo), acceleration (a), and time (t), then use the formula Vf = Vo + at to find Vf. Substitute the known values into the formula to calculate the final velocity.

Explanation:

To solve for final velocity (Vf) in the formula for acceleration, follow these steps:

Identify the initial velocity, Vo, and any other known variables such as acceleration (a) and time (t).

Determine which equation to use, using the relationship that acceleration is the change in velocity over time (Δv/Δt). The formula to calculate final velocity is v = Vo + at.

Substitute the known values into the equation to solve for Vf. For example, with an initial velocity (Vo) of 30.0 km/h (which converts to 8.333 m/s), an unknown final velocity (Vf), and a time (Δt) of 8.00 seconds, the change in velocity (Δv) would be Vf - Vo.

To find Vf, rearrange the equation: Vf = Vo + (a Δt). Insert the known values into the equation and calculate the result.

As an example, if the initial velocity Vo is 0 and the acceleration is 0.40 m/s² over a time of 100 seconds, then the final velocity Vf is calculated as:

Vf = Vo + at = 0 + (0.40 m/s²) (100 s) = 40 m/s.

This procedure will yield the final velocity of an object assuming constant acceleration over the given time period.

Answer:

297 pages

Step-by-step explanation:

Number of pages of longest book = 396 pages

Now we are given that her shortest book has one-fourth as many pages as the longest.

Number of pages of shortest book = [tex]\frac{\text{Number of pages in longest book}}{4}[/tex]

                                            = [tex]\frac{396}{4}[/tex]

                                            = [tex]99[/tex]

Now we are given that the book in the middle of her shelf has three times the number of pages of the shortest book.

So, Number of pages in middle book:

=  [tex]3 \times \text{Number of pages of shortest book}[/tex]

=  [tex]3 \times 99[/tex]

=  [tex]297[/tex]

Hence the middle book have 297 pages.

To determine which system has no solutions, we analyze a set of inequalities to identify the contradictory statement.

To determine which system has no solutions, we need to analyze the given inequalities:

x < 5

x > 1

x < 5

x < 1

x > 5

x > 1

x > 5

x < 1

From the list, the system 'x < 1' and 'x > 1' has no solutions as it creates a contradictory statement.

If EF = 6 and EG = 21 what is the value of FG?

The answer is 15.

The farmer owns 8 10/45 acres of land.

To find out how many acres of land the farmer owns, we need to multiply the number of sections by the size of each section. The size of each section is 2 1/5 acres.

The number of sections is 4 4/9. Using multiplication, we can calculate the total number of acres:

(2 1/5) x (4 4/9) = 10/5 x 40/9 = 400/45 = 8 10/45

Therefore, the farmer owns 8 10/45 acres of land.

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To find the 60th percentile, multiply 60/100 by the total number of observations plus one, which equates to 4.8. This falls between the 4th and 5th value, and rounding up means the 60th percentile corresponds to the 5th value, which is 47.

To find the value that corresponds to the 60th percentile in the given data set {12, 28, 35, 42, 47, 49, 50}, we need to follow a series of steps.

Therefore, the value corresponding to the 60th percentile in this data set is 47.

Final answer:

The 60th percentile value in the data set {12, 28, 35, 42, 47, 49, 50} is 47, which is the 5th number in the ordered set after calculating the index.

Explanation:

The 60th percentile value in a data set is found by first arranging the data in ascending order, which is already done in the provided data set. Then calculate the index using the formula Index = P(N + 1), where P is the percentile in decimal form and N is the number of data points. In this case, the 60th percentile index for our 7-point data set is 0.60 × (7 + 1) = 4.8. Since the index is not a whole number, we round up to the next whole number, which here would be 5. Therefore, we take the 5th number in the ordered set, which is 47.